Applying velocity to non rigid body object? By math?

Here is a example script to get you started. Attach it to a game object in a new scene and use the arrow keys (assuming that is how you have "Horizontal" and "Vertical" setup) to move the object:

 #pragma strict
 
 public var velocityFactor = 0.01;
 public var dragFactor = 0.975; 
 public var maxVelocity = 1.5;
     
 private var velocity : Vector3 = Vector3.zero;
  
 function FixedUpdate(){
     velocity += Vector3(Input.GetAxis("Horizontal"), 0.0, Input.GetAxis("Vertical")) * velocityFactor;
     velocity *= dragFactor;
     velocity = Vector3.ClampMagnitude(velocity, maxVelocity);
     transform.position += velocity;
 }

Every GameObject has a Transform component which tracks its position, scale, and rotation. It's easy enough to change those values. The harder part is changing them in a way that models the behavior you want.

For example, this would roughly simulate gravity:

 public class Falling : MonoBehaviour {
     public Vector3 constantGravity = Vector3.down * 9.8f;
     public Vector3 initialVelocity = Vector3.zero;
     
     Vector3 velocity;
     
     void Start() {
         velocity = initialVelocity;
     }
     
     void Update() {
         velocity += constantGravity * Time.deltaTime; //v = v0 + at
         transform.position += velocity * Time.deltaTime;
     }
 }

For the effects you described, you can probably get away with tracking acceleration and velocity which change once per frame. If your measurements are in units-per-second, multiplying by Time.deltaTime will keep your math divorced from frame rate.

As you might expect, this sort of thing is much easier if you have a basic understanding of physics and/or calculus.